Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays
Author:
Shigui Ruan
Journal:
Quart. Appl. Math. 59 (2001), 159-173
MSC:
Primary 34K20; Secondary 34K18, 92D25
DOI:
https://doi.org/10.1090/qam/1811101
MathSciNet review:
MR1811101
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The dynamics of delayed systems depend not only on the parameters describing the models but also on the time delays from the feedback. A delay system is absolutely stable if it is asymptotically stable for all values of the delays and conditionally stable if it is asymptotically stable for the delays in some intervals. In the latter case, the system could become unstable when the delays take some critical values and bifurcations may occur. We consider three classes of Kolmogorov-type predator-prey systems with discrete delays and study absolute stability, conditional stability and bifurcation of these systems from a global point of view on both the parameters and delays.
- Margarete Baptistini and Plácido Táboas, On the stability of some exponential polynomials, J. Math. Anal. Appl. 205 (1997), no. 1, 259–272. MR 1426993, DOI https://doi.org/10.1006/jmaa.1996.5152
- M. S. Bartlett, On theoretical models for competitive and predatory biological systems, Biometrika 44 (1957), 27–42. MR 86727, DOI https://doi.org/10.1093/biomet/44.1-2.27
- Richard Bellman and Kenneth L. Cooke, Differential-difference equations, Academic Press, New York-London, 1963. MR 0147745
- Edoardo Beretta and Yang Kuang, Convergence results in a well-known delayed predator-prey system, J. Math. Anal. Appl. 204 (1996), no. 3, 840–853. MR 1422776, DOI https://doi.org/10.1006/jmaa.1996.0471
- F. G. Boese, Stability criteria for second-order dynamical systems involving several time delays, SIAM J. Math. Anal. 26 (1995), no. 5, 1306–1330. MR 1347422, DOI https://doi.org/10.1137/S0036141091200848
- Fred Brauer, Characteristic return times for harvested population models with time lag, Math. Biosci. 45 (1979), no. 3-4, 295–311. MR 538430, DOI https://doi.org/10.1016/0025-5564%2879%2990064-6
- Fred Brauer, Absolute stability in delay equations, J. Differential Equations 69 (1987), no. 2, 185–191. MR 899158, DOI https://doi.org/10.1016/0022-0396%2887%2990116-1
M. Brelot, Sur le problème biologique héréditaire de deux espéces dévorante et dévoré, Ann. Mat. Pura Appl. 9, 58–74 (1931)
- Yulin Cao and H. I. Freedman, Global attractivity in time-delayed predator-prey systems, J. Austral. Math. Soc. Ser. B 38 (1996), no. 2, 149–162. MR 1414356, DOI https://doi.org/10.1017/S0334270000000540
- Yuan-shun Chin, Unconditional stability of systems with time-lags, Acta Math. Sinica 10 (1960), 125–142 (Chinese, with English summary). MR 114031
- Kenneth L. Cooke and Zvi Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86 (1982), no. 2, 592–627. MR 652197, DOI https://doi.org/10.1016/0022-247X%2882%2990243-8
- Kenneth L. Cooke and Pauline van den Driessche, On zeroes of some transcendental equations, Funkcial. Ekvac. 29 (1986), no. 1, 77–90. MR 865215
- Jim M. Cushing, Integrodifferential equations and delay models in population dynamics, Springer-Verlag, Berlin-New York, 1977. Lecture Notes in Biomathematics, Vol. 20. MR 0496838
- Lo Sheng Dai, Nonconstant periodic solutions in predator-prey systems with continuous time delay, Math. Biosci. 53 (1981), no. 1-2, 149–157. MR 613620, DOI https://doi.org/10.1016/0025-5564%2881%2990044-4
- R. Datko, A procedure for determination of the exponential stability of certain differential-difference equations, Quart. Appl. Math. 36 (1978/79), no. 3, 279–292. MR 508772, DOI https://doi.org/10.1090/S0033-569X-1978-0508772-8
- J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
- A. Farkas, M. Farkas, and G. Szabó, Multiparameter bifurcation diagrams in predator-prey models with time lag, J. Math. Biol. 26 (1988), no. 1, 93–103. MR 929973, DOI https://doi.org/10.1007/BF00280175
- H. I. Freedman and K. Gopalsamy, Nonoccurrence of stability switching in systems with discrete delays, Canad. Math. Bull. 31 (1988), no. 1, 52–58. MR 932613, DOI https://doi.org/10.4153/CMB-1988-008-0
- H. I. Freedman and V. Sree Hari Rao, The trade-off between mutual interference and time lags in predator-prey systems, Bull. Math. Biol. 45 (1983), no. 6, 991–1004. MR 727356, DOI https://doi.org/10.1016/S0092-8240%2883%2980073-1
- H. I. Freedman and V. Sree Hari Rao, Stability criteria for a system involving two time delays, SIAM J. Appl. Math. 46 (1986), no. 4, 552–560. MR 849081, DOI https://doi.org/10.1137/0146037
- Narendra S. Goel, Samaresh C. Maitra, and Elliott W. Montroll, On the Volterra and other nonlinear models of interacting populations, Rev. Modern Phys. 43 (1971), 231–276. MR 0484546, DOI https://doi.org/10.1103/RevModPhys.43.231
- K. Gopalsamy, Harmless delays in model systems, Bull. Math. Biol. 45 (1983), no. 3, 295–309. MR 708998, DOI https://doi.org/10.1016/S0092-8240%2883%2980058-5
- K. Gopalsamy, Delayed responses and stability in two-species systems, J. Austral. Math. Soc. Ser. B 25 (1984), no. 4, 473–500. MR 734969, DOI https://doi.org/10.1017/S0334270000004227
- K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Mathematics and its Applications, vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992. MR 1163190
- Jack K. Hale, Ettore F. Infante, and Fu Shiang Peter Tsen, Stability in linear delay equations, J. Math. Anal. Appl. 105 (1985), no. 2, 533–555. MR 778486, DOI https://doi.org/10.1016/0022-247X%2885%2990068-X
- Jack K. Hale and Sjoerd M. Verduyn Lunel, Introduction to functional-differential equations, Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993. MR 1243878
- Brian D. Hassard, Nicholas D. Kazarinoff, and Yieh Hei Wan, Theory and applications of Hopf bifurcation, London Mathematical Society Lecture Note Series, vol. 41, Cambridge University Press, Cambridge-New York, 1981. MR 603442
- Alan Hastings, Age-dependent predation is not a simple process. I. Continuous time models, Theoret. Population Biol. 23 (1983), no. 3, 347–362. MR 711912, DOI https://doi.org/10.1016/0040-5809%2883%2990023-0
- Alan Hastings, Delays in recruitment at different trophic levels: effects on stability, J. Math. Biol. 21 (1984), no. 1, 35–44. MR 770711, DOI https://doi.org/10.1007/BF00275221
- Xue-zhong He, Stability and delays in a predator-prey system, J. Math. Anal. Appl. 198 (1996), no. 2, 355–370. MR 1376269, DOI https://doi.org/10.1006/jmaa.1996.0087
W. Huang, Algebraic criteria on the stability of the zero solutions of the second order delay differential equations, J. Anhui University, 1–7 (1985)
G. E. Hutchinson, Circular cause systems in ecology, Ann. New York Acad. Sci. 50, 221–246 (1948)
- Yang Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, vol. 191, Academic Press, Inc., Boston, MA, 1993. MR 1218880
- Zhengyi Lu and Wendi Wang, Global stability for two-species Lotka-Volterra systems with delay, J. Math. Anal. Appl. 208 (1997), no. 1, 277–280. MR 1440357, DOI https://doi.org/10.1006/jmaa.1997.5301
- Ma Zhien, Stability of predation models with time delay, Appl. Anal. 22 (1986), no. 3-4, 169–192. MR 860988, DOI https://doi.org/10.1080/00036818608839617
- J. M. Mahaffy, A test for stability of linear differential delay equations, Quart. Appl. Math. 40 (1982/83), no. 2, 193–202. MR 666674, DOI https://doi.org/10.1090/S0033-569X-1982-0666674-3
R. M. May, Time delay versus stability in population models with two and three trophic levels, Ecology 4, 315–325 (1973)
- Norman MacDonald, Time lags in biological models, Lecture Notes in Biomathematics, vol. 27, Springer-Verlag, Berlin-New York, 1978. MR 521439
- Len Nunney, The effect of long time delays in predator-prey systems, Theoret. Population Biol. 27 (1985), no. 2, 202–221. MR 797394, DOI https://doi.org/10.1016/0040-5809%2885%2990010-3
- Len Nunney, Absolute stability in predator-prey models, Theoret. Population Biol. 28 (1985), no. 2, 209–232. MR 809778, DOI https://doi.org/10.1016/0040-5809%2885%2990028-0
S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations, preprint.
- G. Stépán, Great delay in a predator-prey model, Nonlinear Anal. 10 (1986), no. 9, 913–929. MR 856874, DOI https://doi.org/10.1016/0362-546X%2886%2990078-7
V. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie, Gauthier-Villars, Paris, 1931
P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology 38, 136–139 (1957)
- Tao Zhao, Yang Kuang, and H. L. Smith, Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal. 28 (1997), no. 8, 1373–1394. MR 1428657, DOI https://doi.org/10.1016/0362-546X%2895%2900230-S
M. Baptistini and P. Táboas, On the stability of some exponential polynomials, J. Math. Anal. Appl. 205, 259–272 (1997)
M. S. Bartlett, On theoretical models for competitive and predatory biological systems, Biometrika 44, 27–42 (1957)
R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963
E. Beretta and Y. Kuang, Convergence results in a well-known delayed predator-prey system, J. Math. Anal. Appl. 204, 840–853 (1996)
F. G. Boese, Stability criteria for second-order dynamical systems involving several time delays, SIAM J. Math. Anal. 26, 1306–1330 (1995)
F. Brauer, Characteristic return times for harvested population models with time lag, Math. Biosci. 45, 295–311 (1979)
F. Brauer, Absolute stability in delay equations, J. Differential Equations 69, 185–191 (1987)
M. Brelot, Sur le problème biologique héréditaire de deux espéces dévorante et dévoré, Ann. Mat. Pura Appl. 9, 58–74 (1931)
Y. Cao and H. I. Freedman, Global attractivity in time-delayed predator-prey systems, J. Austral. Math. Soc. Ser. B 38, 149–162 (1996)
Y.-S. Chin, Unconditional stability of systems with time-lags, Acta Math. Sinica 1, 125–142 (1960)
K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86, 592–627 (1982)
K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj 29, 77–90 (1986)
J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer-Verlag, Heidelberg, 1977
L. S. Dai, Nonconstant periodic solutions in predator-prey systems with continuous time delay, Math. Biosci. 53, 149–157 (1981)
R. Datko, A procedure for determination of the exponential stability of certain differential difference equations, Quart. Appl. Math. 36, 279–292 (1978)
J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960
A. Farkas, M. Farkas, and G. Szabó, Multiparameter bifurcation diagrams in predator-prey models with time lag, J. Math. Biol. 26, 93–103 (1988)
H. I. Freedman and K. Gopalsamy, Nonoccurrence of stability switching in systems with discrete delays, Canad. Math. Bull. 31, 52–58 (1988)
H. I. Freedman and V. S. H. Rao, The tradeoff between mutual interference and time lags in predator-prey systems, Bull. Math. Biol. 45, 991–1004 (1983)
H. I. Freedman and V. S. H. Rao, Stability criteria for a system involving two time delays, SIAM J. Appl. Anal. 46, 552–560 (1986)
N. S. Goel, S. C. Maitra, and E. W. Montroll, On the Volterra and other nonlinear models of interacting populations, Rev. Modern Phys. 43, 231–276 (1971)
K. Gopalsamy, Harmless delay in model systems, Bull. Math. Biol. 45, 295–309 (1983)
K. Gopalsamy, Delayed responses and stability in two-species systems, J. Austral. Math. Soc. Ser. B 25, 473–500 (1984)
K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992
J. K. Hale, E. F. Infante, and F.-S. P. Tsen, Stability in linear delay equations, J. Math. Anal. Appl. 105, 533–555 (1985)
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993
B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, London, 1981
A. Hastings, Age-dependent predation is not a simple process: I. Continuous time models, Theoret. Pop. Biol. 23, 347–362 (1983)
A. Hastings, Delays in recruitment at different trophic levels: Effects on stability, J. Math. Biol. 21, 35–44 (1984)
X.-Z. He, Stability and delays in a predator-prey system, J. Math. Anal. Appl. 198, 355–370 (1996)
W. Huang, Algebraic criteria on the stability of the zero solutions of the second order delay differential equations, J. Anhui University, 1–7 (1985)
G. E. Hutchinson, Circular cause systems in ecology, Ann. New York Acad. Sci. 50, 221–246 (1948)
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993
Z. Lu and W. Wang, Global stability for two-species Lotka-Volterra systems with delay, J. Math. Anal. Appl. 208, 277–280 (1997)
Z. Ma, Stability of predation models with time delay, Applicable Analysis 22, 169–192 (1986)
J. M. Mahaffy, A test for stability of linear differential delay equations, Quart. Appl. Math. 40, 193–202 (1982)
R. M. May, Time delay versus stability in population models with two and three trophic levels, Ecology 4, 315–325 (1973)
N. MacDonald, Time Lags in Biological Models, Springer-Verlag, Heidelberg, 1978
L. Nunney, The effect of long time delays in predator-prey systems, Theoret. Pop. Biol. 27, 202–221 (1985)
L. Nunney, Absolute stability in predator-prey models, Theoret. Pop. Biol. 28, 209–232 (1985)
S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations, preprint.
G. Stépán, Great delay in a predator-prey model, Nonlinear Analysis 10, 913–929 (1986)
V. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie, Gauthier-Villars, Paris, 1931
P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology 38, 136–139 (1957)
T. Zhao, Y. Kuang, and H. L. Smith, Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Analysis 28, 1373–1394 (1997)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
34K20,
34K18,
92D25
Retrieve articles in all journals
with MSC:
34K20,
34K18,
92D25
Additional Information
Article copyright:
© Copyright 2001
American Mathematical Society